Finite Group Databasebeta

← C2×#2359 #34131

Subgroups — C2×#2359 #34131

Order 256.

1915 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#123C211yesmaximal, characteristic
2C22×C4×D4C211yesmaximal, characteristic
3C22×#117C211yesmaximal, characteristic, complemented
4#2359C2116yesmaximal, direct product factor
5C2×#389C211yesmaximal, characteristic, complemented
6C2×#388C211yesmaximal, characteristic, complemented
7C2×#388C211yesmaximal, characteristic, complemented
8C2×#387C211yesmaximal, characteristic, complemented
9C2×#387C211yesmaximal, characteristic, complemented
10C2×#386C211yesmaximal, characteristic
11C2×#385C211yesmaximal, characteristic, complemented
12C2×#385C211yesmaximal, characteristic, complemented
13C2×C4×C4⋊C4C211yesmaximal, characteristic, complemented
14C2×C4×C22⋊C4C211yesmaximal, characteristic, complemented
15C2×C4×C22⋊C4C211yesmaximal, characteristic, complemented
16C22×#95C211yesmaximal, characteristic, complemented
17C23×D4C2211yescharacteristic
18C24×C4C2211yescharacteristic
19C24×C4C2211yescharacteristic
20C24×C4C2211yescharacteristic
21C24×C4C2211yescharacteristic
22C2×#123C4116yescomplemented
23C2×#123C2218yes—
24C2×C4×D4—28no—
25C2×C4×D4C2218yes—
26C2×#117—28no—
27C2×#117C2218yescomplemented
28C22×C4⋊C4C2211yescharacteristic
29C22×C4⋊C4C2211yescharacteristic
30C22×C4⋊C4C2211yescharacteristic
31C22×C4⋊C4C2211yescharacteristic
32C22×C4⋊C4C2211yescharacteristic
33C22×C4⋊C4C2211yescharacteristic
34C22×C4⋊C4C2211yescharacteristic
35C22×C22⋊C4C2211yescharacteristic
36C22×C22⋊C4C2211yescharacteristic
37C22×C22⋊C4C2211yescharacteristic
38C22×C22⋊C4C2211yescharacteristic
39C22×C22⋊C4C2211yescharacteristic
40C22×C22⋊C4C2211yescharacteristic
41C22×C22⋊C4C2211yescharacteristic
42C22×C22⋊C4C2211yescharacteristic
43C22×C42C2211yescharacteristic
44C22×C42C2211yescharacteristic
45C22×C42C2211yescharacteristic
46C22×C42C2211yescharacteristic
47C22×C42C2211yescharacteristic
48#389C2218yescomplemented
49#388C2218yescomplemented
50#388C2218yescomplemented
51#387C2218yescomplemented
52#387C2218yescomplemented
53#386C2218yes—
54#385C2218yescomplemented
55#385C2218yescomplemented
56C4×C4⋊C4C2218yescomplemented
57C4×C22⋊C4C2218yescomplemented
58C4×C22⋊C4C2218yescomplemented
59C2×#95C2211yescharacteristic
60C2×#95C2211yescharacteristic
61C2×#95C2211yescharacteristic
62C2×#95C2211yescharacteristic
63C2×#95C2211yescharacteristic
64C2×#95C2211yescharacteristic
65C2×#95C2211yescharacteristic
66C2×#95C2211yescharacteristic
67C2×#95C2211yescharacteristic
68C2×#95C2211yescharacteristic
69C2×#95C2218yescomplemented
70C25C2311yescharacteristic
71C25C2311yescharacteristic
72C22×D4—24no—
73C22×D4—24no—
74C22×D4C2×C418yes—
75C22×D4C2314yes—
76C23×C4—24no—
77C23×C4—24no—
78C23×C4—24no—
79C23×C4—24no—
80C23×C4—24no—
81C23×C4—24no—
82C23×C4—24no—
83C23×C4—24no—
84C23×C4—42no—
85C23×C4—42no—
86C23×C4—42no—
87C23×C4C2×C418yescomplemented
88C23×C4C2311yescharacteristic
89C23×C4C2311yescharacteristic
90C23×C4C2311yescharacteristic
91C23×C4C2311yescharacteristic
92C23×C4C2314yes—
93C23×C4C2314yes—
94C23×C4C2314yes—
95C23×C4C2311yescharacteristic
96C23×C4C2311yescharacteristic
97C23×C4C2311yescharacteristic
98C23×C4C2311yescharacteristic
99C23×C4C2311yescharacteristic
100C23×C4C2314yes—
101C23×C4C2311yescharacteristic
102C23×C4C2311yescharacteristic
103C23×C4C2311yescharacteristic
104C23×C4C2311yescharacteristic
105#123C2×C4164yescomplemented
106C4×D4—232no—
107#117—232no—
108C2×C4⋊C4—24no—
109C2×C4⋊C4—24no—
110C2×C4⋊C4—24no—
111C2×C4⋊C4—24no—
112C2×C4⋊C4—42no—
113C2×C4⋊C4C2×C418yescomplemented
114C2×C4⋊C4C2×C418yescomplemented
115C2×C4⋊C4C2314yes—
116C2×C4⋊C4C2314yes—
117C2×C4⋊C4C2314yes—
118C2×C4⋊C4C2314yes—
119C2×C4⋊C4C2314yes—
120C2×C4⋊C4C2314yes—
121C2×C4⋊C4C2314yes—
122C2×C22⋊C4—24no—
123C2×C22⋊C4—24no—
124C2×C22⋊C4—24no—
125C2×C22⋊C4—42no—
126C2×C22⋊C4C2×C418yescomplemented
127C2×C22⋊C4C2×C418yescomplemented
128C2×C22⋊C4C2×C418yescomplemented
129C2×C22⋊C4D418yes—
130C2×C22⋊C4C2314yes—
131C2×C22⋊C4C2314yes—
132C2×C22⋊C4C2314yes—
133C2×C22⋊C4C2314yes—
134C2×C22⋊C4C2314yes—
135C2×C22⋊C4C2314yes—
136C2×C22⋊C4C2314yes—
137C2×C22⋊C4C2314yes—
138C2×C42—24no—
139C2×C42—24no—
140C2×C42—24no—
141C2×C42—24no—
142C2×C42—42no—
143C2×C42C2314yes—
144C2×C42C2314yes—
145C2×C42C2314yes—
146C2×C42C2314yes—
147C2×C42C2314yes—
148#95C2314yes—
149#95C2314yes—
150#95C2314yes—
151#95C2314yes—
152#95C2314yes—
153#95C2314yes—
154#95C2314yes—
155#95C2314yes—
156#95C2314yes—
157#95C2314yes—
158C24—22no—
159C24—22no—
160C24—22no—
161C24—22no—
162C24—22no—
163C24—22no—
164C24—22no—
165C24—22no—
166C24—22no—
167C24—22no—
168C24C22×C414yes—
169C24C22×C414yes—
170C24C2×D414yes—
171C24#4114yes—
172C24C2412yes—
173C24C2412yes—
174C24C2411yescharacteristic
175C2×D4—232no—
176C2×D4—28no—
177C2×D4—28no—
178C2×D4C22×C4116yes—
179C22×C4—232no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—28no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—28no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—28no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—28no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—28no—
215C22×C4—28no—
216C22×C4—28no—
217C22×C4—28no—
218C22×C4—44no—
219C22×C4—44no—
220C22×C4—44no—
221C22×C4—416no—
222C22×C4—416no—
223C22×C4—416no—
224C22×C4C22×C414yes—
225C22×C4C22×C414yes—
226C22×C4C22×C414yes—
227C22×C4C22×C414yes—
228C22×C4C22×C414yes—
229C22×C4C22×C4116yescomplemented
230C22×C4C2×D414yes—
231C22×C4C2×D414yes—
232C22×C4#4114yes—
233C22×C4#4114yes—
234C22×C4C2412yes—
235C22×C4C2412yes—
236C22×C4C2412yes—
237C22×C4C2412yes—
238C22×C4C2412yes—
239C22×C4C2412yes—
240C22×C4C2412yes—
241C22×C4C2412yes—
242C22×C4C2412yes—
243C22×C4C2412yes—
244C22×C4C2412yes—
245C22×C4C2412yes—
246C22×C4C2412yes—
247C4⋊C4—28no—
248C4⋊C4—28no—
249C4⋊C4—28no—
250C4⋊C4—28no—
251C4⋊C4—44no—
252C4⋊C4C22×C4116yescomplemented
253C4⋊C4C22×C4116yescomplemented
254C22⋊C4—28no—
255C22⋊C4—28no—
256C22⋊C4—28no—
257C22⋊C4—44no—
258C22⋊C4C22×C4116yescomplemented
259C22⋊C4C22×C4116yescomplemented
260C22⋊C4C22×C4116yescomplemented
261C22⋊C4C2×D4116yes—
262C42—28no—
263C42—28no—
264C42—28no—
265C42—28no—
266C42—44no—
267C23—28no—
268C23—28no—
269C23—28no—
270C23—28no—
271C23—28no—
272C23—28no—
273C23—22no—
274C23—22no—
275C23—22no—
276C23—22no—
277C23—28no—
278C23—28no—
279C23—28no—
280C23—22no—
281C23—22no—
282C23—22no—
283C23—22no—
284C23—22no—
285C23—22no—
286C23—44no—
287C23—44no—
288C23—44no—
289C23—44no—
290C23C4×D4116yes—
291C23C23×C414yes—
292C23C23×C414yes—
293C23C23×C412yes—
294C23C22×D414yes—
295C23C22×D412yes—
296C23C2×#4114yes—
297C23C2×#4112yes—
298C23C2×#4112yes—
299C23C2×#4112yes—
300C23H(5,𝔽2)12yes—
301C232−1+412yes—
302C23C2511yescharacteristic
303D4—232no—
304C2×C4—28no—
305C2×C4—28no—
306C2×C4—232no—
307C2×C4—28no—
308C2×C4—28no—
309C2×C4—28no—
310C2×C4—28no—
311C2×C4—28no—
312C2×C4—28no—
313C2×C4—28no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—416no—
345C2×C4—416no—
346C2×C4—416no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4C23×C414yes—
352C2×C4C23×C414yes—
353C2×C4C23×C414yes—
354C2×C4C23×C414yes—
355C2×C4C23×C414yes—
356C2×C4C22×D414yes—
357C2×C4C22×D414yes—
358C2×C4C2×#4114yes—
359C2×C4C2×#4114yes—
360C22—232no—
361C22—24no—
362C22—24no—
363C22—24no—
364C22—24no—
365C22—24no—
366C22—24no—
367C22—24no—
368C22—24no—
369C22—24no—
370C22—416no—
371C22—42no—
372C22—42no—
373C22—42no—
374C22—42no—
375C22C2×C4×D418yes—
376C22C2×C4×D414yes—
377C22C4×#4114yes—
378C22#52014yes—
379C22#54614yes—
380C22#54714yes—
381C22#55114yes—
382C22#55214yes—
383C22C24×C411yescharacteristic
384C22C23×D411yescharacteristic
385C22C22×#4111yescharacteristic
386C22C22×#4111yescharacteristic
387C22C22×#4111yescharacteristic
388C22C2×H(5,𝔽2)11yescharacteristic
389C22C2×2−1+411yescharacteristic
390C4—24no—
391C4—24no—
392C4—24no—
393C4—24no—
394C4—24no—
395C4—24no—
396C4—24no—
397C4—24no—
398C4—24no—
399C4—42no—
400C4—42no—
401C4—42no—
402C4—42no—
403C2—28no—
404C2—44no—
405C2#235918yesdirect product factor
406C2C22×C4×D411yescharacteristic
407C2C2×C4×#4111yescharacteristic
408C2C2×#52011yescharacteristic
409C2C2×#54611yescharacteristic
410C2C2×#54711yescharacteristic
411C2C2×#55111yescharacteristic
412C2C2×#55211yescharacteristic
subgroupG/H
center, socleC24C24
commutatorC22C24×C4
Frattini, cosocleC23C25
prime radicalC23×D4C22
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C23 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. C22×#117C2semidirect
4. #2359C2direct (N × H)
5. C2×#389C2semidirect
6. C2×#388C2semidirect
7. C2×#388C2semidirect
8. C2×#387C2semidirect
9. C2×#387C2semidirect
11. C2×#385C2semidirect
12. C2×#385C2semidirect
13. C2×C4×C4⋊C4C2semidirect
14. C2×C4×C22⋊C4C2semidirect
15. C2×C4×C22⋊C4C2semidirect
16. C22×#95C2semidirect
22. C2×#123C4semidirect
27. C2×#117C22semidirect
48. #389C22semidirect
49. #388C22semidirect
50. #388C22semidirect
51. #387C22semidirect
52. #387C22semidirect
54. #385C22semidirect
55. #385C22semidirect
56. C4×C4⋊C4C22semidirect
57. C4×C22⋊C4C22semidirect
58. C4×C22⋊C4C22semidirect
69. C2×#95C22semidirect
87. C23×C4C2×C4semidirect
105. #123C2×C4semidirect
113. C2×C4⋊C4C2×C4semidirect
114. C2×C4⋊C4C2×C4semidirect
126. C2×C22⋊C4C2×C4semidirect
127. C2×C22⋊C4C2×C4semidirect
128. C2×C22⋊C4C2×C4semidirect
229. C22×C4C22×C4semidirect
252. C4⋊C4C22×C4semidirect
253. C4⋊C4C22×C4semidirect
258. C22⋊C4C22×C4semidirect
259. C22⋊C4C22×C4semidirect
260. C22⋊C4C22×C4semidirect
405. C2#2359direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C22×#1232
C22×C4×D42
C22×#1172
#23592
C2×#3892
C2×#3882
C2×#3872
C2×#3862
C2×#3852
C2×C4×C4⋊C42
C2×C4×C22⋊C42
C22×#952